# T1 space

A topological space $(X,\tau)$ is said to be $T_{1}$ (or said to hold the $T_{1}$ axiom) if for all distinct points $x,y\in X$ ($x\neq y$), there exists an open set $U\in\tau$ such that $x\in U$ and $y\notin U$.

A space being $T_{1}$ is equivalent to the following statements:

• For every $x\in X$, the set $\{x\}$ is closed.

• Every subset of $X$ is equal to the intersection of all the open sets that contain it.

• Distinct points are separated.

 Title T1 space Canonical name T1Space Date of creation 2013-03-22 12:18:14 Last modified on 2013-03-22 12:18:14 Owner drini (3) Last modified by drini (3) Numerical id 10 Author drini (3) Entry type Definition Classification msc 54D10 Synonym T1 Related topic T0Space Related topic T2Space Related topic T3Space Related topic RegularSpace Related topic ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA Related topic SierpinskiSpace Related topic PropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2